
\documentclass[aps,floatfix,prd,twocolumn]{revtex4}
%\documentclass[aps,floatfix,prd,twocolumn]{revtex4}
\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{ amssymb }

\voffset 1.0cm

\begin{document}


\title{Probing the Halo Environments of Globular Clusters with White Dwarfs}
\author{T. Hurst, A. Zentner, C. Badenes and A. Natarajan}
\affiliation{
Department of Physics and Astronomy,
University of Pittsburgh,
Pittsburgh, PA 15260,
USA \& The PITTsburgh Particle Physics and Cosmology Center (PITT-PACC).}

\date{\today}

\begin{abstract}
White dwarfs should capture dark matter as they orbit their host halos.  If dark matter is self-interacting, then this dark matter will annihilate, heating the core of the star and preventing the white dwarf from cooling.  Thus, given a dark matter model and the white dwarf cooling sequence of a globular cluster, one can constrain the dark matter density in the cluster.  In this paper, we show that if the claims of a low mass DM particle from CRESST, DAMA, CDMS-Si, and CoGent are correct, then observations of NGC 6397 limit the fraction of dark matter in that cluster to be $\lesssim 10^{-3}$-$10^{-5}$.  This is an improvement over the existing constraint by 3-5 orders of magnitude.

\end{abstract}

\maketitle

\section{Introduction}

In the current Cosmological paradigm, $\sim 23\%$ of the energy of the Universe is in the form of Dark Matter (DM).  The exact nature of this DM is currently unknown, but the most popular particle candidate is the Weakly Interacting Massive Particle (WIMP).  WIMPs are a compelling candidate because, among other reasons, one can recover the proper relic abundance simply by assuming a weak-scale interaction cross-section and because WIMP candidates arise naturally in various extensions to the Standard Model (SM) of particle physics.  As such, detecting and identifying the DM WIMP is of utmost importance to both cosmology and particle physics.  Accordingly, a great deal of resources have been brought to bear on this problem.

One avenue of inquiry into the nature of DM is the so-called direct detection experiment.  These experiments seek to identify DM by observing scattering events in a detector on Earth.  Some of the ongoing experiments in this field are CoGent \cite{CoGent}, CDMS \cite{CDMS}, CRESST \cite{CRESST} and DAMA \cite{DAMA}.  While recent results from the LUX experiment \cite{LUX} cast doubt on their results, currently, there is a tenous agreement emerging between these experiments of a DM particle with a mass of $M_\chi \sim$ 5-20 GeV and cross-section for collision off of a proton $\sigma_{\chi\text{p}} \sim 10^{-41}$ cm$^2$ (see e.g. figure 4 in \cite{CDMS}).  If we can indeed identify the DM particle through these experiments and measure it's properties with sufficient accuracy, then we can perform what some authors have referred to as WIMP astronomy (e.g. \cite{Peter}).  The idea being that with the properties of DM known, we can use indirect detections of DM, say from DM annihilation products, to deduce the structure (and sub-structure) of a DM halo.  

One potential application of WIMP astronomy is the following:  It is well known that stars should capture DM as they orbit their host halo (see e.g. the review \cite{Jungman}).  Therefore, if DM is self-annihilating there should be some heating from DM annihilation in the cores of stars.  In particular, White Dwarfs (WDs) should have some heating in their cores which prevents them from cooling below a certain temperature \cite{Hooper}.  Hence, given a DM model, we can constrain the local DM density around a WD by assuming all of its luminosity is from DM annihilation.  Alternatively, given information on the DM content of an object (from dynamical observations, say) we could constrain the DM model.  

In the present work we suggest looking at WDs in Globular Clusters (GCs) in order to place limits on the DM content of the clusters and consider NGC 6397 as a particular exampe.  We will focus our attention on DM with the properties necessary to explain the recent results from CoGent, CDMS, CRESST and DAMA.

GCs are interesting in this context because they are the only known structures that show no evidence for DM.  For instance, \cite{Conroy} found that for NGC 2419 and MGC1 $M_{\text{DM}}/M_* < 1$ where $M_{\text{DM}}$ is the mass in DM of the cluster and $M_*$ is the stellar mass of the cluster.  Subsequently \cite{Shin} found a similar result for NGC 6397.  In the present work we show how WIMP astronomy can improve this constraint by several orders of magnitude.

\section{Methods}

To calculate the the rate at which a WD will capture DM we use the result (A.16) from \cite{Zentner}
\begin{equation}
C_{\text{c}} = \sqrt{\frac{3}{2}}\frac{\rho_\chi}{M_\chi}\sigma v_{\text{esc}}(R)\frac{v_{\text{esc}}(R)}{\bar{v}}N\langle\hat{\phi}\rangle\frac{\text{erf}(\eta)}{\eta},
\end{equation}
where $\rho_\chi$ is the local DM density, $R$ is the radius of the WD, $v_{\text{esc}}(R)$ is the escape speed at the surface of the star, $\bar{v}$ is the dispersion of the DM velocity profile (assumed to be Maxwell-Boltzmann), $\langle\hat{\phi}\rangle$ is an average potential well depth for stellar nucleons, and $\eta$ is the ratio of the star's velocity through its halo to the local velocity dispersion of the halo.  Since we are interested in capture off of nucleons, $\sigma = \sigma_{\text{N}}$ and $N$ is the number of a given nuclear species in the star.  The total capture rate, is the sum over all species. 

Here we consider stable DM of mass $M_\chi > 3$ GeV and neglect self-capture off of already bound DM particles.  Then the number of DM particles captured within the star, $N_\chi$, is governed by the differential equation 
\begin{equation}  
\frac{dN_\chi}{dt} = C_{\text{c}} - C_{\text{a}}N_\chi^2,
\end{equation}
where $C_{\text{a}}$ is twice the annihilation rate (because each annihilation eliminates 2 particles).  The solution to this equation for homogeneous initial conditions is
\begin{equation}
N_\chi = \sqrt{\frac{C_{\text{c}}}{C_{\text{a}}}}\tanh(\sqrt{C_{\text{c}}C_{\text{a}}}t).
\end{equation}
There is an equilibration timescale $\tau_{\text{eq}} = 1/\sqrt{C_{\text{c}}C_{\text{a}}}$, such that for $t > \tau_{\text{eq}}$, $N_\chi$ approaches a steady state solution $N_{\chi,\text{eq}} = \sqrt{C_{\text{c}}/C_{\text{a}}}$.  Then the annihilation rate within our WD will be
\begin{equation}
\Gamma_{\text{a}} = \frac{1}{2}C_{\text{a}}N_{\chi,\text{eq}}^2 = \frac{1}{2}C_{\text{c}},
\end{equation}
because there are $N_{\chi,\text{eq}}^2$/2 distint pairs of DM particles within the star.  As in \cite{Hooper} this should produce a luminosity 
\begin{equation}
L_\chi \approx \Gamma_{\text{a}}M_\chi,
\end{equation}
and of course the luminosity of the white dwarf is
\begin{equation}
L_{\text{WD}} = 4\pi R^2\sigma_{\text{SB}}T^4,
\end{equation}
where $\sigma_{\text{SB}}$ is the Stefan-Boltzmann constant and $T$ is the effective temperature of the WD.  As the WD cools, it will eventually reach some critical temperature $T_{\text{c}}$ such that $L_\chi = L_{\text{WD}}$ and the WD can no longer cool.  Hence, we can use WDs to put an upper bound on the local DM density of their host halos.  

Now, the cross-section for scattering off of a given nucleus is approximately
\begin{equation}
\sigma_{\text{N}} = \sigma_{\chi\text{p}}A^2\frac{M_\chi^2M_{\text{N}}^2}{{(M_\chi+M_{\text{N}})}^2}\frac{{(M_\chi+m_{\text{p}})}^2}{M_\chi^2m_{\text{p}}^2},
\end{equation}
where $A$ is the atomic mass number of the nucleus of interest, $M_{\text{N}}$ is the mass of the nucleus, and $m_{\text{p}}$ is the proton mass.  For simplicity we will consider a WD made entirely from Carbon and Oxygen.  Then we can write the composition of the WD as
\begin{equation}
N = N_{\text{C}}  + N_{\text{O}} = f_{\text{C}}\frac{M_{\text{WD}}}{M_{\text{C}}} + f_{\text{O}}\frac{M_{\text{WD}}}{M_{\text{O}}},
\end{equation}
where f$_{\text{C}}$ and f$_{\text{O}}$ are the fractions of Carbon and Oxygen and obviously f$_{\text{C}} + f_{\text{O}} = 1$.  (Our results are not sensitive to the WD composition.)  For a WD we have that $\langle\hat{\phi}\rangle \approx 1.5$, and of course, we also have that
\begin{equation}
v_{\text{esc}}(R) = \sqrt{\frac{2GM_{\text{WD}}}{R}},
\end{equation}
where G is Newton's constant.
\section{Results}
Let us consider the case of NGC 6397, which has mass $M_{6397} = 1.1 \pm 0.1$ x $10^5$ M$_\odot$ \cite{Heyl} and half-light radius $R_{\text{HL}}$ = 2.2 pc \cite{Harris}.  We can estimate the average density of NGC 6397 as
\begin{equation}
\bar{\rho} = \frac{M_{6397}}{\frac{8}{3}\pi R_{\text{HL}}^3} \approx 4.7 \text{ x } 10^4 \text{ GeV/cm}^3.
\end{equation}
Let us define the fraction of DM in the cluster as f$_{\text{DM}} = \rho_\chi/\bar{\rho}$.

The cooling sequence of WDs in NGC 6397 has been measured very well with deep Hubble Space Telescope imaging.  As discussed in appendix A of \cite{Hansen} the truncation of the WD cooling sequence occurs at an absolute magnitude $M = 15.15 \pm 0.15$.  The best fit model for the WD is a mass at truncation $\sim 0.6$ M$_\odot$ which corresponds to a cooling age $\sim 11.0$ $\pm$ 0.5 Gyr.  Using the cooling models of \cite{Fontaine} we find $T_{\text{eff}} \sim$ 3500-3700 K.

We now consider f$_{\text{DM}}$ as a function of $\sigma_{\chi\text{p}}$ with best fit values for $M_\chi$ and $\sigma_{\chi\text{p}}$ taken from \cite{CDMS}, \cite{CoGent}, \cite{CRESST} and \cite{DAMA}.  Figure (\ref{fig:rhovsigma}) shows that if we take the DM model seriously, then CDMS, CoGent, and CRESST requre f$_{\text{DM}} \lesssim 10^{-3}$ while DAMA implies f$_{\text{DM}} \lesssim 10^{-5}$.

\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{rhovsigma}
\caption{The DM fraction vs. proton scattering cross-section for NGC 6397:  The solid lines are the constraints for the best fit values for $M_\chi$ from the direct detection experiments and span the range 3500-3700 K.  (The constraints from all of the experiments lie within this band).  The vertical dashed lines denote the best fit cross-section for the corresponding experiment.  The black dotted line shows the existing constraint, f$_{\text{DM}} < 1$.}
\label{fig:rhovsigma}
\end{figure}
Figure (\ref{fig:rhovmass}) shows the corresponding constraints for f$_{\text{DM}}$ as a function of $M_\chi$.  While in figure (\ref{fig:mvsigma}) we consider $\sigma_{\chi\text{p}}$ as a function of $M_\chi$ for fixed f$_{\text{DM}}$.

\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{rhovmass}
\caption{The DM fraction vs. DM particle mass:  The solid lines are the constraints for the best fit values for $\sigma_{\chi\text{p}}$ from the direct detection experiments and span the range 3500-3700 K.  The vertical dashed lines denote the best fit mass for the corresponding experiment.}
\label{fig:rhovmass}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{mvsigma}
\caption{The DM-proton scattering cross-section vs. DM particle mass:  The solid lines are the constraints for the corresponding value of  f$_{\text{DM}}$ and span the range 3500-3700 K.  The black diamonds denote the best fit values for DAMA (top), CoGent, CRESST, and CDMS (Bottom).}
\label{fig:mvsigma}
\end{figure}

\section{Conclusions}
White dwarfs should accumulate dark matter in their cores as they orbit their host halos.  This DM should in turn annihilate, heating the core and preventing the WD from cooling beyond some temperature.  Therefore, we can use the coolest white dwarf in a globular cluster to put an upper limit on the dark matter density in the cluster.  In particular, we have seen that if the low mass WIMP detections from CoGent, Dama, CRESST, and CDMS are confirmed, observations of the WD cooling sequence in NGC 6397 limit f$_{\text{DM}} \lesssim 10^{-3}$-$10^{-5}$.  This is a significant improvement over existing constraints.  

\acknowledgments
We would like to thank Brad Hansen and Jason Kalirai for their help in locating and understanding the observational data for NGC 6397.

\begin{thebibliography}{99}

\bibitem{CDMS}
R. Agnese {\it et al.}, arXiv:hep-ex/1304.4279v3 (2013).

\bibitem{Cline} 
J. M. Cline {\it et al.}, arXiv:hep-ph/1207.3039v2 (2012).

\bibitem{CoGent}
C. E. Aalseth {\it et al.}, arXiv:astro-ph/1002.4703v2 (2010).

\bibitem{Conroy}
C. Conroy {\it et al.}, AP J 741, 72 (2011).

\bibitem{CRESST}
G. Angloher  {\it et al.}, arXiv:astro-ph/1109.0702v1 (2011).

\bibitem{DAMA}
C. Savage  {\it et al.}, arXiv:astro-ph/0901.2713v2 (2009).

\bibitem{Fontaine}
G. Fontaine {\it et al.}, PASP 113, 409 (2001).

\bibitem{LUX}
D. S. Akerib {\it et al.}, arXiv:astro-ph/1310.8214v1 (2013).

\bibitem{Hansen}
B. M. S. Hansen {\it et al.}, arXiv:astro-ph/0701738v2 (2007).

\bibitem{Harris}
W. E. Harris Ap J, 112, 1487 (1996).

\bibitem{Heyl}
J. S. Heyl {\it et al.}, arXiv:astro-ph/1210.0826v2 (2012).

\bibitem{Hooper}
D. Hooper {\it et al.}, arXiv:hep-ph/1002.0005v1 (2010).

\bibitem{Jungman}
G. Jungman {\it et al.}, arXiv:hep-ph/9506380 (1996).

\bibitem{Peter}
A. H. G. Peter, arXiv:astro-ph/1103.5145v2 (2011).

\bibitem{Shin}
J. Shin {\it et al.}, JKAS 46, 173 (2013).

\bibitem{Zentner}
A. Zentner, arXiv:astro-ph/0907.3448v2 (2009).

\end{thebibliography}

\end{document}



